Method for the interferometric measurement of non-rotationally symmetric wavefront errors

ABSTRACT

The method serves for the interferometric measurement of non-rotationally symmetric wavefront errors on a specimen . The specimen is brought into a number of rotational positions, at least one measurement result being determined in each of the rotational positions and a mathematical evaluation of all measurement results is performed. The measurement results (M 1  . . . M m ; N 1  . . . N n ) of each of the measurement series (M, N) are determined respectively in mutually equidistant rotational positions of the specimen . The measurement results (M 1  . . . M m , N 1  . . . N n ) of each of the at least two measurement series (M, N) are evaluated independently of one another for non-rotationally symmetric wavefront errors (&lt;W&gt; m , &lt;W&gt; n ) on the specimen, and a difference is computationally rotated m or n times and the results averaged out. At least one of the wavefront errors (&lt;W&gt; m , &lt;W&gt; n ) is corrected with the result (&lt;&lt;W&gt; m -&lt;W&gt; n &gt; m  or &lt;&lt;W&gt; m -&lt;W&gt; n &gt; n ) averaged in this way.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The invention relates to a method for the interferometric measurement ofnon-rotationally symmetric wavefront errors on a specimen brought into anumber of rotational positions, at least one measurement result beingdetermined in each of the rotational positions, and a concludingmathematical evaluation of all measurement results being performed.

2. Description of the Related Art

A fundamental requirement of each precision measurement techniqueconsists in determining the measurement instrument error. Provided forthis purpose are standards whose error contributions are either known orcan be separated from the measuring instrument error by suitablemeasuring methods. The error of the measuring instrument is determinedfrom the respective measurement and the error of the respectivestandard. In many cases, however, no such standards are available, forwhich reason it is necessary to depend on methods for separating thecontributions of measuring instrument errors and testing surface.

In the optical surface measuring technique, testing surfaces aremeasured with the aid of an interferometric measuring method. For thispurpose, a shape-matched wavefront is aligned with the specimen, and theshape of the reflected wavefront is measured. In addition to the surfaceshape of the testing surface, errors of the measuring instrument arealso found impressed onto this wavefront.

As regards the general prior art, reference is made to the publicationsby R. Freimann, B. Dörband, F. Höller: “Absolute measurement ofNon-Comatic Aspheric Surface Errors”, Optics Communication, 161,106-114, 1999; and C. J. Evans, R. N. Kestner: “Test Optics ErrorRemoval”, Applied Optics, Vol. 35, 7, 1996; the JP 8-233552 and the U.S.Pat. No. 5,982,490.

The possibility in principle of using two independent measurement seriesis pointed out specifically on page 1018, 2nd paragraph, in thepublication by Evans and Kestner listed above, although no specificmethod is named. Moreover, it is pointed out explicitly at this juncturethat such a method requires specific assumptions on the errors to beexpected, and a (mathematically complicated) fitting of the measureddata.

SUMMARY OF THE INVENTION

Therefore the object of the invention is to provide a method for theinterferometric measurement of the non-rotationally symmetric wavefronterrors of optical surfaces in reflection and/or optical elements intransmission, which provides a higher accuracy than the generally knownrotational position test with a comparable number of partialmeasurements, or which, with a significantly lower number of partialmeasurements, provides a comparable accuracy to the rotational positiontest.

According to the invention, this object is achieved by virtue of thefact that the measurement is carried out in at least two measurementseries, the measurement results of each of the measurement series beingdetermined in respectively mutually equidistant rotational positions ofthe specimen, each of the measurement series comprising a specificnumber n, m of measurements, and m and n being natural and mutuallycoprime numbers, the measurement results of each of the at least twomeasurement series being evaluated independently of one another fornon-rotationally symmetric wavefront errors on the specimen, thedifference of the at least two non-rotationally symmetric wavefronterrors being formed, whereupon the difference that is formed iscomputationally rotated m times and the results are averaged out, andwhereupon the wavefront errors is corrected with the result averaged inthis way.

According to the invention, this object is also achieved. In this wayinstead of m times the difference is computationally rotated n times.

Through the use of at least two independent measurement series, each ofwhich has a number of equidistant measurement points, it is herepossible to achieve a significant improvement in the measurementaccuracy and/or a reduction in the number of individual measurementpoints required.

To that end, each of the at least two measurement series has a specificnumber of measurement results, for example m and n in the case of twomeasurement series. Through these m+n measured rotational positions anda corresponding mathematical evaluation, it is now possible to achievethe situation that all non-rotationally symmetric errors of the specimenwith the exception of the orders k·m·n can be established absolutely. Inorder to achieve the maximum achievable accuracy with a minimum numberof measurements, the number of individual measurement results m and nmust be mutually coprime.

Certainly, measurement methods with two measurement series M, N or threemeasurement series M, N, O are primarily to be regarded as an expedientapplication of the method according to the invention, although four,five, six or more measurement series are in principle also conceivable.In the case of two measurement series M, N which, for example, consistof a combination of m=5 measurements in the first measurement series Mand n=7 measurements in the second measurement series N, a higheraccuracy is achieved than in the case of a 12-position test, forexample. With the proposed 5+7 measurements, one of the measurementresults is furthermore duplicated, so that only 11 measurements in totalneed to be carried out here. With a corresponding 12-position testaccording to the prior art, all non-rotationally symmetric errors of thespecimen up to the order k·12 can be established. With the cited exampleof the method according to the invention as a 5+7-position test, allerrors with the exception of the orders k·5·7=k·35 can already beestablished with 11 measurements in total, i.e. one measurement less.

If this rotational position test is extended to three measurementseries, for example with a 3+5+7-position test, then all errors up tothe order k·3·5·7=k·105 can be established. In this test, only 13measurements are necessary instead of the theoretical 15 measurements,since one of the measurements occurs three times. Hence, withapproximately the same number of individual measurement results to berecorded, the accuracy of the measurement can be increased significantlywith the method according to the invention, the entire circumference ofthe specimen furthermore being covered.

As an alternative to this, it would naturally also be possible to reducethe amount of measurement time through a corresponding reduction of theindividual measurement results, for example a 3+4-position test which,since one measurement occurs twice, requires only 6 measurements. Anaccuracy up to errors of the order k-12 could likewise be achievedthereby, as in the previously known 12-position test. The decisiveadvantage here, however, is that the number of individual measurements,and therefore the required measurement time, for achieving a comparableaccuracy is reduced by 50 percent without this requiring specific andcomplicated algorithms and software.

According to the invention, the object is also achieved by virtue of thefact that the specimen is measured in a first rotational position,whereupon, in a second step, for determining the odd orders of thenon-rotationally symmetric wavefront components, the specimen is rotatedby 180° and is measured, and, in k further steps (k=1, 2, 3 . . . ), fordetermining the non-rotationally symmetric wavefront components of theeven orders u*2^(k) the specimen is rotated by 360/2^(k+1) and ismeasured, u being a natural odd number and k any natural number.

The object is also achieved wherein after the measurement of thespecimen in a first rotational position the non-rotationally symmetricwavefront components of the even orders u*2^(k) are determined in asecond step, and the odd orders of the non-rotationally symmetricwavefront components are determined in a third step.

Here, the method according to the invention for determiningnon-rotational symmetric surface components of a specimen ischaracterized by an exponential dependence of the last still correctlytransmitted azimuthal order on the number of the partial measurementsrequired therefore. In order, for example, to determine allnon-rotationally symmetric wavefront orders up to and including order255, only nine partial measurements are now required. If, for example,the number of partial measurements is raised to twelve, it is possiblethereby to extract all non-rotationally symmetric wavefront componentsup to and including order 2047.

The number of partial measurements can be substantially reduced, and themeasuring times for the absolute measurements of non-rotationallysymmetric wavefront components can thereby be shortened by means of thismethod according to the invention. In this way, there is also areduction in the influence of drift effects, and this leads to animprovement in absolute accuracy. A further advantage of the shortenedoverall measuring time is the saving in time resulting there from.

The method “works” with differences between interferometric individualmeasurements. In this case, the interferometer errors are not averagedout by summation over various rotational positions, but are subtractedwith pixel accuracy. The application of this method is particularlyadvantageous if the interferometer error is greater than the error ofthe specimen. Any disturbing influence on the result can be ruled outhere because the interferometer error is eliminated before thenon-rotationally symmetric wavefront components of the specimen isascertained.

Exemplary embodiments of the invention are explained in principle belowin more detail with the aid of the drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a highly schematized representation of a measurement setupwhich is possible in principle;

FIG. 2 shows the position of measurement points on a specimen using theexample of a 3+5-position test; and

FIG. 3 shows the measuring positions of a specimen for a 2^(n)rotationally averaging measurement.

DETAILED DESCRIPTION

FIG. 1 shows a highly schematized outline representation of a setup forcarrying out the method for the interferometric measurement ofnon-rotationally symmetric wavefront errors on a specimen 1. Aninterferometric measuring instrument 2, indicated here in outline, isused for this. The interferometric measuring instrument 2 isschematically represented as a Michelson interferometer 2, although itmay in principle be any other conceivable type of interferometricmeasuring instrument with the known facilities of splitting up the lightpaths, through semi-silvered mirrors, optical fibers, couplers or thelike with open light paths or non-open light paths (e.g. fibre opticinterferometer).

Besides a light source 3, the interferometric measuring instrument 2 hasa reference element 4 and an instrument 5 for recording the interferencepattern that is created. The instrument 5 is coupled to an evaluationunit 6 which, for example, is designed as an electronic data processingunit and in which the required mathematical evaluation of allmeasurement results can be carried out.

An arrow indicates the required relative rotational movement R betweenthe interferometric measuring instrument 2 and the specimen 1, whichneeds to be carried out between the recording of the individualmeasurement results. The variant selected in this case for the schematicrepresentation is the supposedly simple one in which the specimen 1 isrotated relative to the interferometric measuring instrument 2. It is,of course, also conceivable for the interferometric measuring instrument2 to be rotated about the specimen 1.

FIG. 2 represents the position of measurement points using the exampleof m=3 and n=5 individual measurements of two measurement series M, N,which, for each measurement series M, N, are arranged distributed atequidistant spacings or angular positions over the entire circumferenceof the specimen 1.

In principle, the sequence in which the individual measurement values M₁. . . M_(m), N₁ . . . N_(n) are recorded is unimportant in this case. Itis, however, expedient for stability, and therefore the measurementreliability to be achieved in the setup, if the rotational direction ofthe relative rotational movement R between the interferometric measuringinstrument 2 and the specimen 1 is maintained throughout the entiremeasurement.

In the exemplary embodiment represented, it could therefore be expedientto carry out the measurements in the sequence M₁, M₂, M₃, N₂, N₃, N₄,N₅. The measurement N₁ can in this case be omitted, since it isprecisely this measurement result which is already known from themeasurement M₁. By maintaining the rotational direction and therotational angles, which are respectively equidistant within ameasurement series, it is hence possible to achieve a very highstability of the setup in the method, which represents a goodprerequisite for qualitative high-value measurement results. With thisprocedure, in which first of all the m equidistant spacings of onemeasurement series M and then the n or (n−1) equidistant spacings of theother measurement series N are addressed, however, the situation arisesthat the specimen 1 needs to be rotated completely at least two timesrelative to the interferometric measuring instrument 2. Nevertheless,good reproducibility can be achieved here because of the angularspacings that can respectively be set equidistantly.

In principle, however, it is also conceivable to carry out themeasurement method with only one rotation of the specimen 1, in whichcase the individual measurement positions according to the example inFIG. 2 are then addressed in the sequence M₁, N₂, M₂, N₃, N₄, M₃, N₅.Since the individual measuring points of each of the measurement seriesM, N need to be mutually equidistant and measurement points from the twomeasurement series M, N are now determined in mixed fashion, therequired mechanical accuracy of the setup for carrying out the relativerotational movements R is in this case somewhat higher, since themeasurement points to be successively recorded no longer lie at amutually equidistant spacing from each other here, and this situation ismore difficult to implement with the required accuracy andreproducibility.

The achievable measurement accuracy of the described method and adescription of the possibilities for evaluating the measurement resultsthat are obtained will be given below with the aid of mathematicalalgorithms.

Basically, in any interferometric testing of the specimen 1 forwavefront errors, the measured wavefrontW=P+Tcan be represented as a sum of the wavefront errors of the specimen:P=P _(r) +P _(nr)and of the interferometer:T=T _(r) +T _(nr).

In this case, P_(r) and T_(r) respectively denote the rotationallysymmetric component) P_(nr) and T_(nr) of the non-rotationally symmetriccomponent of the measured wavefront. For representation of the wavefrontin a sum notation, the following is hence obtained:W=T _(r) +T _(nr) +P _(r) +P _(nr)  (1).

Now if, in a measurement according to the prior art, m rotationalpositions at an azimuthal spacing of 360°/m are measured and averagedout, then all non-rotationally symmetric errors of the specimen 1 dropout, with the exception of the orders k·m·e (with k=1, 2, 3, . . . ),which gives:<W> _(m) =T _(r) +T _(nr) +P _(r) +P _(nr) ^(kme ()2).

In order to determine the error of the specimen 1, equation (2) can nowbe subtracted from equation (1), and all non-rotationally symmetricerrors of the specimen 1 up to the orders k·m·e are obtained with:W−<W> _(m) =P _(nr) −P _(nr) ^(kme)  (3).If e.g. the rotational position test with m=12 rotational positions isassumed, then this means that all non-rotationally symmetric errors ofthe specimen 1 up to the orders 12, 24, 36, . . . are obtained. Thismeans that the errors in the vicinity of the 12^(th) periodicity,24^(th) periodicity, 36^(th) periodicity etc. cannot be recorded withthis measurement method.

If, in the method, a further measurement series with n rotationalpositions is now carried out in a similar way to the measurementdescribed in the introduction, which is known from the prior art, thenthe following is obtained in a similar way to that described above:<W> _(n) =T _(r) +T _(nr) +P _(r) +P _(nr) ^(kne)  (4)andW−<W> _(m) =P _(nr) −P _(nr) ^(kne)  (5).

If the results of equations (2) and (4) are now subtracted from oneanother, the error contributions of the orders k·m·e and k·n·e areobtained, with the exception of the orders k·m·n·e that are actuallycontained in both of the measurement series M, N, with:<W> _(m) −<W> _(n) =P _(nr) ^(kme) −P _(nr) ^(kne)  (6).A prerequisite for this, of course, is that m and n are coprime numbers.

If this wavefront described by equation (6) is now computationallyrotated m times, for example with the aid of a corresponding softwareprogram, and the results are averaged out, then the k·n·e terms dropout.

If the result obtained is added to equation (3), then thenon-rotationally symmetric errors of the specimen 1, with the exceptionof the k·m·n·e terms, are obtained with:W _(m) =W−<W> _(m) +<<W> _(n) −<W> _(n)>_(m) =P _(nr) −P _(nr)^(kmne)  (7).

In a similar way to this, it is naturally also possible to calculate theresult for n rotational positions, in order to use it for furtheraveraging or for analysis.

This method hence offers the opportunity, merely with a number ofmeasurements m+n, to determine in absolute terms all non-rotationallysymmetric errors of the specimen 1, with the exception of the ordersk-n-m. A combination of 5+7 measurements, which corresponds to 11individual measurements because of the one measurement that occurstwice, is therefore much more accurate than the rotational position testwith, for example, 12 measurement points.

As an alternative to this, of course, it is also possible to achievesimilar accuracies to the aforementioned 12-position test with fewerindividual measurements, for example, 3+4 measurements. Since, however,only 6 measurement points are needed for this, one of the theoretical 7measurement points occurring twice, the required measurement time, orthe required measurement work, can be halved.

An alternative formulation will be described below, which is obtainedwhen the individual measurements according to equation (1) arecomputationally rotated back to a common azimuthal position before beingdetermined. In this case, the non-rotationally symmetric interferometererrors, with the exception of the orders k·m·e, are removed:<W> _(m) =T _(r) +T _(nr) ^(kme) +P _(r) +P _(nr)  (8).

If the mean radial profile of the wavefrontW _(RP) =T _(r) +P _(r)  (9)is now computationally established, and is subtracted from the wavefrontaccording to equation (8), then all non-rotationally symmetric errors ofthe specimen 1 and, in addition, also the non-rotationally symmetricerrors of the interferometer 2 of the orders k·m·e, are obtained with:<W> _(m) −W _(RP) =P _(nr) +T _(nr) ^(kme)  (10).If the same procedure is performed in a similar way with n rotationalpositions, then this gives:<W> _(n) =T _(r) +P _(r) +P _(nr) +T _(nr) ^(kne)  (11)and<W> _(n) −W _(RP) =P _(nr) +T _(nr) ^(kne)  (12).

If the two results of equations (8) and (11) are now subtracted from oneanother, the error contributions of the orders k·m·e and k·n·e, with theexception of the orders k-m-n-e, are obtained so long as m and n areagain coprime numbers, since these are contained in both of themeasurement series M, N, as:<W> _(m) −<W> _(n) =T _(nr) ^(kme) −T _(nr) ^(kne)  (13).

If the wavefront described by equation (13) is now computationallyrotated m times and averaged out, the k·n·e terms again drop out. Onecould say that the determined error is again subjected to a rotationalposition test, albeit on a purely mathematical or virtual basis.

Subtracting the result from equation (10) gives, with:W _(m) =<W> _(m) −W _(RP) <<W> _(m) −<W> _(n)>_(m) =P _(nr) +T^(kmne)  (14)

The non-rotationally symmetric errors of the specimen, including thek·m·n·e terms of the interferometer error. Here again, the result for nrotational positions can be calculated in a similar way to this, inorder to use it for further averaging or for analysis.

The error can, of course, be further minimized by additional rotationalpositions. For instance, even higher accuracies can be achieved withthree measurement series M, N, O. With a 3+5+7-position test, i.e. withm=3, n=5 and o=7 individual measurements, all errors up to the ordersk·105 can be determined. Since three of the individual measurementsoccur twice, 13 measurement points in total are sufficient to achievethe corresponding accuracy. Here as well, it is again assumed that thenumbers m, n, o are coprime natural numbers.

If n+m measurements are now combined, as indicated above, then thenon-rotationally symmetric errors of the specimen 1, with the exceptionof the k·m·n·e terms, are obtained in a similar way to equation (7)with:W _(M) =W−<W> _(m) +<<W> _(m) −<W> _(n)>_(m) =P _(nr) −P _(nr)^(kmne)  (15).

Furthermore, the non-rotationally symmetric errors of the specimen 1with the exception of the k·n·o·e terms, are obtained from thecombination of the measurements n +o:W _(o) =W −<W _(o) >+<<W> _(o) −<W> _(n)>_(o) =P _(nr) −P _(nr)^(knoe)  (16).

If the two equations (15) and (16) are now subtracted from one another,the error contributions of the orders k·m·n·e and k·n·o·e, with theexception of the orders k-m-n-o-e since these are actually contained inboth results, are obtained with:W _(o) −W _(M) =P _(nr) ^(knme) −P _(nr) ^(knoe)  (17).

If this wavefront described by equation (17) is now also computationallyrotated m times and averaged out, the k·n·o·e terms drop out. By addingthe result to equation (15), the non-rotationally symmetric errors ofthe specimen 1, with the exception of the k·m·n·o·e terms, are obtainedin a similar way to equation (7), with:W _(mn) =W−<W> _(m) +<<W> _(m) −<W> _(n)>_(m) +<W _(o) −W _(M) >=P _(nr)−P _(nr) ^(kmoe)  (18).

Here again, the results for n rotational positions can naturally becalculated and used for further averaging or for analysis, as alreadymentioned above.

Again here as well, algorithms which take into account thenon-rotationally symmetric interferometer errors with three measurementseries can be achieved by computational rotation to a common azimuthalposition, in a similar way to the procedure with two measurement seriesM, N. Corresponding algorithms for measurement methods having more thanthree measurement series M, N, O, . . . are likewise obtained in asimilar way to the possibilities described above.

FIG. 3 illustrates the rotational angles of the partial measurementscovered for the 2^(n) rotational averaging method, only two partialmeasurements being required to determine the non-rotationally symmetricwavefront components of the specimen 1.

The measuring accuracy to be targeted for the described measuringmethod, and a description of the possibilities for evaluating thetargeted measurement results are to be represented below with the aid ofmathematical algorithms here, as well.

The wavefront W₀ determined according to FIG. 1 by means of theinterferometric measuring instrument 2 can be split into the errors P ofthe specimen 1 and into the errors T of the interferometric measuringinstrument 2, a distinction being made between rotationally symmetricand non-rotationally symmetric wavefront components in accordance withthe following formula (19):W _(o) =P _(r) +P _(nr)(θ)+T _(r) +T _(nr)  (19).

In this case, P_(r) and T_(r) respectively denote the rotationallysymmetric component, and P_(nr) and T_(nr) the non-rotationallysymmetric component of the measured wavefront W_(o).

The specimen 1 is now measured in a first rotational position, forexample at θ=0°, and thereafter rotated by an arbitrary angle α andmeasured once again. In order to illuminate the rotationally symmetricwavefront components of the specimen 1, as well as the rotationalsymmetric and non-rotational symmetric interferometer errors, themeasured value W_(α) that was determined at an angle α is subtractedfrom the measured value W₀ that was determined for 0°. The result ofthis is that only the non-rotationally symmetric specimen error remains,in accordance with equation (20):W ₀ −W _(α) =P _(nr)(θ)−P _(nr)(θ+α)  (20).

All rotational symmetric errors of the specimen 1 and of theinterferometric measuring instrument 2 thus drop out, as do thenon-rotational symmetric errors of the interferometric measuringinstrument 2. Consequently, in this way the interferometer errors areeliminated by a design of the rotationally symmetric surface componentsof the specimen 1 on the basis of forming a difference.

On the basis of this principle, the odd orders of the non-rotationallysymmetric wavefront components P_(nr)(θ) of the specimen 1 aredetermined in a further step. For this purpose, the specimen 1 ismeasured at a rotational angle of θ=0°, the result being a first partialmeasurement W₀. Thereafter, the specimen 1 is rotated by 180° (θ=180°),and a second measurement ω₁₈₀ is carried out. In this case, therotational angle θ=180° corresponds to half a period of the first order(single wave property), as a result of which the sign of the oddnon-rotationally symmetric wavefront components of the specimen 1 isinverted, and the following equation is obtained:P _(nr)(1*θ+180)=−P _(nr)(1*θ)  (21).

The number 1 in front of the rotational angle θ in equation (21) standsfor the single wave property, it also being possible for the number 1 tobe replaced by an arbitrary odd number.

If the difference of the partial measurements of the specimen 1 at 0°and at 180° is now formed, the following result:W ₀ −W ₁₈₀=2*P _(nr)(1*θ)  (23)

Is obtained with the aid ofW ₀ −W ₁₈₀ =P _(nr)(1*θ)−P _(nr)(1*θ+180)  (22)that is to say a doubling of this wavefront order.

However, this property holds not only for the single wave property asjust set forth by equations (21) to (23) but also for all the other oddorders of the wavefront with P*(q+1/2)=180°, P corresponding to theperiod=360°/n, and q being a whole natural number. The propertytherefore also holds for the 3-wave property, 5-wave property, 7-waveproperty, etc.

The following two cases are to be distinguished as a function of theorder n of the non-rotationally symmetric wavefront:P _(nr)(n*θ+180)=−P _(nr)(n*θ), when n is an odd natural number, andP _(nr)(n*θ+180)=P _(nr)(n*θ), when n is an even natural number.  (24).

The result for the differenceP _(nr)(n*θ)−P _(nr)(n*θ+180)  (25)of the equations (24) is therefore:P _(nr)(n*θ)−P _(nr)(n*θ+180)=2* P _(nr)(n*θ), when n is an odd naturalnumber, andP _(nr)(n*θ)−P _(nr)(n*θ+180)=0, when n is an even natural number  (26),the non-rotationally symmetric wavefront order P_(nr)(n*θ) being doubledwhen an odd natural number is used for n. P_(nr)(n*θ) vanishes in thecase of an even natural number n.

The difference of the partial measurements W₀ and W₁₈₀:P1(n*θ)≡1/2*(W ₀ −W ₁₈₀)=1/2*(P _(nr)(n*θ)−P _(nr)(n*θ+180))  (27)therefore includes all odd orders of the non-rotationally symmetricwavefront components of the specimen 1. Consequently, all odd orders(single wave property, 3-wave property, 5-wave property, 7-waveproperty, . . . ) of the non-rotationally symmetric wavefront componentsare determined. Not determined, however, are all even orders of thenon-rotationally symmetric wavefront components of the specimen 1 thatare to be determined by means of the following equations.

All the even orders 2^(k) with k=1, 2, 3, 4, 5, . . . of the wavefrontcomponents of the specimen 1 will be determined using the proceduredescribed below. For this purpose, further measurements W_(360/( 2)^(k+1) ₎ are carried out after rotating the specimen 1 by 360/2^(k+1)°in accordance with FIG. 3, and subtracted again from the measurement at0°:W ₀ −W _(360/(2) ^(k+1) ₎ =P _(nr)(n*θ)−P _(nr)(n*θ+360/2^(k+1))  (28).

Depending on which (even) order of the non-rotationally symmetricwavefront that is to be determined, it is necessary to select a specificwhole number for k, for example k=1 must be set in order to determinethe two-wave property. An example for determining the two-wave propertywill be explained in more detail later.

The difference in accordance with equation (28) contains all the evenorders of the non-rotationally symmetric wavefront components of thespecimen 1 with P*q≠360/2^(k+1), P corresponding to the periodlength=360/n°, and q being a whole natural number. It holds for the evenorders or wave properties with P*(q+1/2)=360/2^(k+10) that:P _(nr)(n*θ+360/2^(k+1))=−P_(nr)(n*θ)  (29)

Three cases are to be distinguished thereby:1. P _(nr)(n*θ)−P_(nr)(n*θ+360/2^(k+1))=2*P(n*θ), for n=u*2^(k) whereu=1, 3, 5, . . . odd number2. P _(nr)(n*θ)−P _(nr)(n*θ+360/2^(k+1))=0, for n=g*2k with g=2, 4, 6, .. . even number3. P _(nr)(n*θ)−P _(nr)(n*θ+360/2^(k+1))< >0 for all remaining values ofn.  (30)

If the differential wavefrontDk(θ)≡1/2*(P _(nr)(n*θ)−P _(nr)(n*+360/2^(k+1)))=1/2*(W ₀ −W₃₆₀/(2^(k+1)))  (31)described by equation (31) is now rotated further computationally(2^(k)−1) times by the angle increment 360/2^(k+1)° with the aid of anappropriate software program, for example, and averaged arithmeticallyover all the wavefronts thus obtained, including the unrotatedwavefront, it is only whole multiples of the 2^(k) wave property thatremain in the resultant wavefront:Pk(θ)=(1/2^(k))*(Dk(θ)+Dk(θ+360/2^(k))+Dk(θ+2*360/2^(k))+ . . .+Dk(θ+(2^(k)−1),*360/2^(k))  (32).

The difference, averaged over 2^(k) rotational positions of theinterferometer measurements W₀ and W_(360/(2) ^(k+1) ₎ thus contains allthe even orders u*2^(k) with u=1, 3, 5, 7, . . . (odd number) of thenon-rotationally symmetric wavefront components of the specimen 1.

If k=1, 2, 3, 4, 5, . . . , is selected, the result is the 2*u, 4*u,8*u, 16*u, 32*u . . . order or wave property of the non-rotationallysymmetric wavefront components with u=1, 3, 5, 7, . . . odd. Thefollowing total result PM(θ) in accordance with equation (33) resultsfrom adding up all the rotationally averaging partial results of therespective even orders according to the equation (32) plus the oddorders P1(θ) according to equation (27):PM(θ)=P1(θ)+P2(θ)+P4(θ)+P8(θ)+ . . . +Pk _((θ))  (33).

As an example, equations (28) to (32) are to be used below to determinethe two-wave property plus their odd multiples.

The first step in this is to measure the specimen 1 at 0°. A furthermeasurement is carried out after rotating the specimen 1 by 90°, thismeasurement result being subtracted from the measurement result at 0°:W ₀ −W ₉₀ =P _(nr)(n*θ)−P _(nr)(n*θ+90)  (34).

This differential result therefore again includes all the waveproperties with P*q≠90°, with P corresponding to the periodlength=360°/n, and q being a whole number. For wave properties withP*(q+1/2)=90°, it holds further that:P _(nr)(n*θ+90)=−P _(nr)(n*θ)  (35).

Three cases are therefore again to be distinguished after forming thedifference of equations (34) and (35):1. P _(nr)(n*θ)−P _(nr)(n*θ+90)=2*P _(nr)(n*θ) for n=2*u with u=1, 3, 5. . . odd number2. P _(nr)(n*θ)−P _(nr)(n*θ+90)=0, for n=2*g with g=2, 4, 6, . . . evennumber3. P _(nr)(n*θ)−P _(nr)(n*θ+90)< >0 for all remaining values of n.  (36)

Case 1 therefore relates to the orders 2*u with u=1, 3, 5, 7, . . . ,which means that the equation in accordance with case 1 includes theorders 2, 6, 10, . . . The case 2 includes the orders 2*g with g=2, 4,6, . . . , that is to say the equation in accordance with case 2includes the orders 4, 8, 12, .. . . . Case 3 includes all the otherorders that are not covered by cases 1 and 2.

The difference of the measurement results W₀ for 0° and W₉₀ for 90°therefore includes a superposition of all the three cases. In order nowto average the difference out of case 3, that is to say to obtain thepure 2*u wave property, the latter is isolated by rotational averaging,as already mentioned. Thus, averaging is carried out computationallyover two rotational positions usingD2(θ)≡(−1/2*(P _(nr)(n*θ)−P _(nr)(n*θ+90))=1/2*(W ₀ −W ₉₀ )  (37),and this means that the 2-wave property and multiples thereof aredetermined:P2(θ)≡1/2*(D2(θ)+D2(θ+180))  (38).

The result of equation (38) P2(θ), which represents the difference ofthe interferometer measurements W₀ and W₉₀ degrees rotationally averagedover two rotational positions therefore includes the wave properties 2,6, 10, . . . or 2*u with u=1, 3, 5, 7, . . . odd.

The further even orders (4-wave property, 6-wave property, 8-waveproperty, . . . ) of the non-rotationally symmetric wavefront componentsof the specimen 1 can also be determined in accordance with therepresentation of the example for determining the two-wave property.Adding up these results yields the result for all the even orders of thenon-rotationally symmetric wavefront components of the specimen 1. Thecomplete non-rotationally symmetric wavefront is obtained by adding theodd orders according to equation (27) for this partial result inaccordance with equation (33).

The following individual measurements are to be executed depending onthe maximum number of measurements M:

Measurements for

-   -   1. 360/2⁰=0    -   2. 360/2¹=180°→u*1 wave property (u=1, 3, 5, . . . odd)    -   3. 360/2²=90°→u*2 wave property    -   4. 360/2³=45°→u*4 wave property    -   5. 360/2⁴=22.5°→u*8 wave property

Mth measurement for 360/2^(M−1)°→u*2^(M−2) wave property

However, it is also possible to determine the even orders of thenon-rotationally symmetric wavefront components of the specimen 1 aftereliminating the interferometer errors and before determining the oddorders of the non-rotationally symmetric wavefront components of thespecimen 1. Procedure for this purpose can be similar to thedetermination of the orders as respectively described.

By comparison with the known rotational averaging method, this methodoffers the possibility of detecting more orders of the non-rotationallysymmetric wavefront components of specimen 1 with a smaller number ofmeasurements. For example, all the orders up to the azimuthal order 15can be determined or detected in absolute terms with only five partialmeasurements. In order, for example, to average over all the orders upto 255, only nine individual measurements are required, the result beinga substantially more accurate measurement result than delivered by theknown rotationally averaging method for the same number of partialmeasurements.

Of course, it is also possible as an alternative to carry out more orfewer individual measurements.

1. A method for the interferometric measurement of non-rotationallysymmetric wavefront components of order u*2^(k) with k being a naturalnumber including zero and u being a natural odd number on a specimenbrought into two rotational positions, at least one measurement resultbeing determined in both rotational positions, whereas in a first step ameasurement is taken in a first rotational position, whereupon in asecond step the specimen is rotated by 360/2^((k+1)) degrees andmeasured, whereas the difference of both measurements is divided by 2and averaged with (2^(k)−1) data maps, obtained by rotating the samedifference computationally by 1 *360/2^(k), 2*360/2^(k), . . . ,(2^(k)−1)*360/2^(k) degrees with the average result containing all nonrotational wavefront components of the specimen with orders u*2^(k). 2.The method as claimed in claim 1, applied for k=0, 1, 2, . . . K,wherein the K results are added together to obtain all non rotationalwavefront components of the specimen up to orders 2^(K)−1 completely. 3.The method as claimed in claim 1, wherein an interferometric absolutemeasurement is carried out.